US10613496B2 - Support structure constrained topology optimization for additive manufacturing - Google Patents
Support structure constrained topology optimization for additive manufacturing Download PDFInfo
- Publication number
- US10613496B2 US10613496B2 US15/269,264 US201615269264A US10613496B2 US 10613496 B2 US10613496 B2 US 10613496B2 US 201615269264 A US201615269264 A US 201615269264A US 10613496 B2 US10613496 B2 US 10613496B2
- Authority
- US
- United States
- Prior art keywords
- topological
- design
- support
- volume
- unconstrained
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active, expires
Links
Images
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B17/00—Systems involving the use of models or simulators of said systems
- G05B17/02—Systems involving the use of models or simulators of said systems electric
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B29—WORKING OF PLASTICS; WORKING OF SUBSTANCES IN A PLASTIC STATE IN GENERAL
- B29C—SHAPING OR JOINING OF PLASTICS; SHAPING OF MATERIAL IN A PLASTIC STATE, NOT OTHERWISE PROVIDED FOR; AFTER-TREATMENT OF THE SHAPED PRODUCTS, e.g. REPAIRING
- B29C64/00—Additive manufacturing, i.e. manufacturing of three-dimensional [3D] objects by additive deposition, additive agglomeration or additive layering, e.g. by 3D printing, stereolithography or selective laser sintering
- B29C64/30—Auxiliary operations or equipment
- B29C64/386—Data acquisition or data processing for additive manufacturing
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B29—WORKING OF PLASTICS; WORKING OF SUBSTANCES IN A PLASTIC STATE IN GENERAL
- B29C—SHAPING OR JOINING OF PLASTICS; SHAPING OF MATERIAL IN A PLASTIC STATE, NOT OTHERWISE PROVIDED FOR; AFTER-TREATMENT OF THE SHAPED PRODUCTS, e.g. REPAIRING
- B29C64/00—Additive manufacturing, i.e. manufacturing of three-dimensional [3D] objects by additive deposition, additive agglomeration or additive layering, e.g. by 3D printing, stereolithography or selective laser sintering
- B29C64/40—Structures for supporting 3D objects during manufacture and intended to be sacrificed after completion thereof
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B33—ADDITIVE MANUFACTURING TECHNOLOGY
- B33Y—ADDITIVE MANUFACTURING, i.e. MANUFACTURING OF THREE-DIMENSIONAL [3-D] OBJECTS BY ADDITIVE DEPOSITION, ADDITIVE AGGLOMERATION OR ADDITIVE LAYERING, e.g. BY 3-D PRINTING, STEREOLITHOGRAPHY OR SELECTIVE LASER SINTERING
- B33Y50/00—Data acquisition or data processing for additive manufacturing
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/49—Nc machine tool, till multiple
- G05B2219/49023—3-D printing, layer of powder, add drops of binder in layer, new powder
Definitions
- This disclosure relates generally to methods of optimizing designs, and more specifically to methods for designing structures that are optimized for manufacture with additive manufacturing processes.
- AM processes are processes for fabricating parts through material addition. Specifically, AM devices manufacture three-dimensional objects by adding layer-upon-layer of material in the “build direction” (e.g., from the bottom to the top of the object). The growing interest in AM stems from its ability to fabricate highly complex parts with relative ease. However, structures built with AM must observe certain limitations of the AM processes. Polymer AM processes work with melted, partially melted, and/or amorphous materials, and unsolidified portions of layers can droop or creep where there is no underlying material providing support. The same “overhanging” portions can be damaged by burning during metal AM processes. Thus, overhanging portions of the structure require support structures to hold the overhanging portions in place during manufacture. These support structures are “sacrificial”—they are made of the same material as the structure being manufactured, and are removed after fabrication.
- Support structures directly add to the build-time and material cost.
- Material costs can be substantial in AM; for example, the largest percentage cost for metal AM, besides the machine cost that is amortized, is material cost (18%).
- support structures can be hard to remove (and sometimes even inaccessible), leading to the post-fabrication (clean-up) cost.
- Post-fabrication costs make-up for about 8% of AM product cost.
- Topology optimization represents a class of computational methods for designing lightweight, high-performance structures. After several years of intensive research, it has emerged as a powerful design tool, and is deployed in optimization of aircraft components, spacecraft modules, automobile components, cast components, compliant mechanisms, etc.
- the overarching goal of TO is to start with a given design that meets specifications for rigidity, load bearing, force resistance, etc., and reduce it to an optimized design that is lighter in weight and uses the least amount of material while meeting the same specifications.
- Designs stemming from TO are geometrically complex, and therefore hard to manufacture using traditional processes, but these designs can often be additively manufactured. Also, since fabrication cost in AM is proportional to the material used, light-weight topology optimized designs are particularly relevant in AM.
- the purpose of this disclosure is to provide a TO methodology for limiting the support structure volume, thereby leading to designs that are AM friendly.
- the present invention overcomes the aforementioned drawbacks by providing systems and processes for additive manufacturing using a topology optimization (TO) framework that generates designs that have significantly reduced support structure requirements during manufacture.
- the TO framework may be implemented in design software, such as computer-aided drafting software, to speed up the design and engineering processes and avoid manual iterative design processes.
- the disclosure provides a method for optimizing an object for additive manufacturing, the object having a first volume of material.
- the method includes: receiving electronic data comprising a first design of the object; receiving a support constraint parameter having a value between zero and one; determining a first support volume of a first number of support structures required to support the object during the additive manufacturing, in a build direction, of the object from the first design; performing a first topological optimization of the first design, the first topological optimization being unconstrained as to support volume, to produce a first unconstrained optimized design, the object in the first unconstrained optimized design comprising a first fractional volume of material that is less than the first volume of material; determining a first unconstrained support volume of a second number of support structures required to support the object during the additive manufacturing, in the build direction, of the object from the first unconstrained optimized design; computing a first topological sensitivity, for a performance of the object, to each of one or more topological changes between the first design and the first unconstrained optimized design; computing a
- Computing the second topological sensitivity may include, for each of the one or more topological changes between the first design and the first unconstrained optimized design, determining a corresponding change to the support structure volume at each point of a plurality of points within the first design that have a corresponding surface normal disposed at an angle from the build direction that exceeds a threshold angle.
- Computing the second topological sensitivity may further include smoothing the second topological sensitivity based on a horizontal overhang distance of each overhang of one or more overhangs in the first design.
- Computing the first augmented topological field may further include combining a first sensitivity field corresponding to the first topological sensitivity with a second sensitivity field corresponding to the second topological sensitivity according to an augmented Lagrangian method to produce the first augmented topological field.
- the method may further include: receiving a target fractional volume that is less than the first fractional volume; performing the first topological optimization of the first unconstrained optimized design to produce a second unconstrained optimized design, the object in the first unconstrained optimized design comprising the target fractional volume of material; determining a second unconstrained support volume of a third number of support structures required to support the object during the additive manufacturing, in the build direction, of the object from the second unconstrained optimized design; computing a third topological sensitivity, for a performance of the object, of the first intermediate design to each of one or more proposed topological changes; computing a fourth topological sensitivity, for a support structure volume required to perform the additive manufacturing of the object in the build direction, of the first intermediate design to each of the one or more proposed top
- the present disclosure provides a computing device that includes memory storing device logic, and a processor in communication with the memory and executing the device logic to: receive an initial design of an object, the object having an initial volume of material in the initial design; and iterate a topological optimization of the initial design to produce a plurality of iterative designs of the object, the plurality of iterative designs including a final optimized design in which the object comprises a final volume of material that is a target fraction of the initial volume of material, wherein each of the iterative designs has a corresponding support volume of support structures required to support the object during additive manufacturing of the object from the iterative design, the corresponding support volume constrained according to a support constraint parameter.
- the present disclosure provides a method for optimizing an object for additive manufacturing, the method including: receiving an initial design of an object, the object having an initial volume of material in the initial design; and, iterating a topological optimization of the initial design to produce a plurality of iterative designs of the object, the plurality of iterative designs including a final optimized design in which the object comprises a final volume of material that is a target fraction of the initial volume of material, wherein each of the iterative designs has a corresponding support volume of support structures required to support the object during additive manufacturing of the object from the iterative design, the corresponding support volume constrained according to a support constraint parameter.
- FIG. 1 is a diagram of an example system configured to transform an initial design into an optimized design according to a support volume sensitive topological optimization framework, in accordance with the present disclosure.
- FIG. 2 is a graphic diagram of a topographical optimization of a triangular bracket, the graph showing the relative compliance change over multiple volume-reducing iterations.
- FIG. 3A is a diagram showing the identification of support structures and support volumes for an initial design.
- FIG. 3B is a graphic diagram of the topographical optimization of FIG. 2 further showing expected support volumes for each illustrated iteration, the graph showing the support volume change over multiple volume-reducing iterations.
- FIG. 4A is a diagram of an initial design for an L-shaped bracket.
- FIG. 4B is a diagram of an initial design for the L-shaped bracket of FIG. 4A with a hypothetical hole added to the interior.
- FIG. 4C is a computer-simulated diagram of a topological sensitivity field for the object design of FIG. 4B .
- FIG. 5A is a diagram of an exemplary optimized topology of the design of FIG. 4A .
- FIGS. 5B and 5C are computer-simulated diagrams of a topological sensitivity field having no support volume constraints.
- FIGS. 6A and 6B are diagrams illustrating different points of perturbation of the topology of FIG. 3A .
- FIG. 7A is a diagram of an exemplary optimized topology of the design of FIG. 4A .
- FIGS. 7B and 7C are computer-simulated diagrams of a support volume topological sensitivity field.
- FIGS. 8A and 8B are computer-simulated diagrams of an augmented topological field combining the topological sensitivity fields of FIGS. 5B-C and 7 B-C.
- FIG. 9 is a flowchart of a method of optimizing an object using a support volume sensitive topographical optimization framework.
- FIG. 10A is a diagram of a Messerschmidt-Bölkow-Blohm (MBB) beam.
- FIG. 10B is a computer-simulated diagram of an unconstrained topographical optimization of the MBB beam.
- FIGS. 10C-E are computer-simulated diagrams of support volume sensitive topographical optimizations of the MBB beam using different support constraints.
- FIG. 11 is a graphic diagram of an iterative, support volume sensitive topographical optimization of the triangular bracket of FIG. 2 , the graph showing a comparison of relative compliance over multiple volume-reducing iterations of the support volume sensitive topographical optimization and the unconstrained topographical optimization of FIG. 2 .
- FIG. 12 is a graph of support volume over multiple volume-reducing iterations of the unconstrained topographical optimization of FIG. 2 and the support volume sensitive topographical optimization of FIG. 11 .
- FIG. 13 is a side perspective view of a pole bracket.
- FIG. 14 is a computer-simulated diagram of an unconstrained topographical optimization of the pole bracket of FIG. 13 .
- FIG. 15 is a computer-simulated diagram of a support volume sensitive topographical optimization of the pole bracket of FIG. 13 .
- FIG. 16 is a graph of support volume over multiple volume-reducing iterations of the unconstrained topographical optimization of FIG. 14 and the support volume sensitive topographical optimization of FIG. 15 .
- FIG. 17 is a graph of relative compliance over multiple volume-reducing iterations of the unconstrained topographical optimization of FIG. 14 and the support volume sensitive topographical optimization of FIG. 15 .
- FIG. 18A is a side perspective view of an automotive part.
- FIG. 18B is a top view of the automotive part of FIG. 18A .
- FIG. 19 is a computer-simulated diagram of an unconstrained topographical optimization of the automotive part of FIG. 18A , with an additive manufacturing build direction along the negative Z axis.
- FIG. 20 is a computer-simulated diagram of a support volume sensitive topographical optimization of the automotive part of FIG. 18A , with an additive manufacturing build direction along the negative Z axis.
- FIG. 21 is a computer-simulated diagram of an unconstrained topographical optimization of the automotive part of FIG. 18A , with an additive manufacturing build direction along the positive Y axis.
- FIG. 22 is a computer-simulated diagram of a support volume sensitive topographical optimization of the automotive part of FIG. 18A , with an additive manufacturing build direction along the positive Y axis.
- FIG. 23 is a computer-simulated diagram of an unconstrained topographical optimization of the automotive part of FIG. 18A , with an additive manufacturing build direction along the positive X axis.
- FIG. 24 is a computer-simulated diagram of a support volume sensitive topographical optimization of the automotive part of FIG. 18A , with an additive manufacturing build direction along the positive X axis.
- Described here are systems and computer-implemented methods for generating designs for additive manufacturing (AM) that are topologically optimized according to a topological optimization (TO) process that maximizes performance, subject to support structure constraints.
- AM additive manufacturing
- TO topological optimization
- Example design results and descriptions of fused deposition models are provided to demonstrate the robustness and efficiency of the disclosed systems and processes.
- the systems and methods can be implemented as an enhancement to existing computer-aided drafting (CAD) software to speed up the design and engineering process, which is typically done manually and iteratively.
- CAD computer-aided drafting
- the systems and methods can also result in better optimization than the manual approach.
- FIG. 1 illustrates an exemplary system for optimizing a design of an object according to a support volume sensitive TO framework.
- a computing device 100 includes a processor 102 that executes device logic 104 within the processor 102 or contained in memory 106 of the computing device 100 .
- the device logic 104 configures the processor 102 to perform the processes described herein.
- the computing device 100 may be a server computer or a system of interconnected server computers, such as a web server, application server, application platform, virtual server, cloud data server, and the like, or a personal computer, laptop computer, tablet computer, e-reader, smartphone, personal data assistant, microconsole, industrial automation system, or similar computing device having, as the processor 102 , a central processing unit (CPU), microprocessor, or other suitable processor.
- CPU central processing unit
- the device logic 104 and/or memory 106 may store program instructions and other data for a computer-aided drafting (CAD) program, or another suitable program, for creating, modifying, exporting, and performing other processes on data (e.g., files, database records, data streams, etc.) representing two- and/or three-dimensional designs of objects that can be fabricated by AM processes.
- CAD computer-aided drafting
- the program instructions and other data for performing the processes herein may cooperate with the CAD program.
- the processor 102 receives, as input, an initial object design 110 .
- the initial object design 110 may be input by a user of an interface 108 , which may be presented to a user on the computing device 100 or on another device, such as a drafting computer.
- the interface 108 may be presented on a display of the user device via a dedicated software application (e.g., a CAD program), an internet browser or other web application, or another suitable application in which the interface 108 is a component, such as in a web dashboard or other administration tool.
- the interface 108 may be configured to prompt the user to provide the initial object design 110 , and may present and facilitate one or more options for doing so.
- the interface 108 may prompt the user to select a file for upload.
- the interface 108 may further prompt the user to enter other data used in the present processes, such as the control parameter ⁇ for determining a support volume constraint, as described below.
- the processor 102 executes the device logic 104 to apply an iterative optimization process 120 to transform the initial object design 110 into an optimized design 130 .
- the optimized design 130 is topologically optimized for performance, i.e., an object manufactured by AM processes from the optimized design 130 performs substantially the same functions as an object manufactured from the initial object design 110 .
- the optimization is further constrained to minimize the total volume of support structures (e.g., support structures 132 ) needed during fabrication of the corresponding object by AM processes.
- the processor 102 may store the optimized design 130 (and any intermediate designs), such as in memory 106 , and/or may export the optimized design 130 to another system, such as an AM device.
- FIG. 2 illustrates a PareTO optimization of an exemplary triangular three-hole bracket 200 , where the two left side holes 202 , 204 are fixed and the right side hole 206 is subject to a downward unit load.
- the progression of the optimization process in PareTO reduces the amount of material needed, from a beginning volume of 1.0 to a volume fraction of 0.5.
- the optimization generates multiple topologies that lie on the Pareto curve (Pareto tracing); illustrated are the eighth iteration topology 210 , the 17th iteration topology 220 , and the 21st iteration topology 230 .
- the generation of multiple topologies plays an important role in the proposed method for constraining the support structure volume.
- Support structure generation in AM is based on the “overhang concept,” which states that if the angle between the boundary normal and the build direction exceeds a certain threshold, then support structures are needed at that point.
- the angle ⁇ between the build direction and the normal N of the unsupported boundary 302 is subtended. Boundary points with the angle ⁇ greater than a threshold, such as 135 degrees, are considered overhanging and require support.
- a threshold such as 135 degrees
- Support structures may terminate at a support platform 330 or at any opposing non-overhanging point on a boundary of the design 300 .
- the union of all such support structures results in a support volume, which is the sum of the shaded areas 320 A-D.
- the fill-ratio, i.e., material density, of support structures is typically less than that of the primary design.
- Support volume may be enclosed between an overhanging surface and an opposing surface, as illustrated by support volume 320 D of FIG. 3A .
- both surfaces must be penalized, such as by moving them closer to each other.
- By penalizing the overhanging surface only half the problem is addressed.
- a simple integral of the support length over the boundary may be multiplied by a suitable fill ratio:
- the exact value of the fill ratio is not critical; it can be assumed to be 0.5, without a loss in generality.
- FIG. 3B illustrates the same PareTO optimization of the triangular bracket 200 as in FIG. 2 , which has no constraints on the support volume (referred to herein as an “unconstrained topological optimization”).
- the necessary support structures 350 , 360 , 370 , 380 are depicted.
- Each iteration of the optimized bracket has a support volume S unc. ( ⁇ ) at its respective intermediate fraction ⁇ .
- the support volume curve is, in general, non-smooth, unlike the compliance curve in FIG. 2 .
- the present TO framework may impose an absolute support volume constraint S ⁇ S max , where S is the total support volume of the optimized design and S max is an upper limit of the total support volume, selected by the designer.
- S is the total support volume of the optimized design
- S max is an upper limit of the total support volume, selected by the designer.
- the absolute constraint will not produce a design with the optimal support volume, and it places an unreasonable burden on the designer to arrive at an absolute value for the upper limit a priori.
- relative upper bound constraints may be imposed, using the PareTO method of generating multiple topologies for various volume fractions to store reference support volumes S( ⁇ ).
- the reference support volumes may be generated according to a relative support volume constraint, S( ⁇ ) ⁇ S unc. ( ⁇ ), where ⁇ is a user-defined control parameter and (0 ⁇ 1). That is, the desired support volume should be less than the unconstrained support volume by a factor of ⁇ .
- the relative support volume constraint may be a ‘soft’ constraint, i.e., the constraint is used to prioritize the solutions within the feasible space, rather than limiting this space.
- the parameter (0 ⁇ 1) may be used to strike a balance between performance and AM costs.
- FIG. 4A presents a first object design 400 that represents a structural topology in the design space ⁇ 0 described above
- FIG. 4B presents a second object design 410 that is the first object design 400 modified to include a small hypothetical hole 412 in the topology.
- Topological sensitivity is the rate of performance change of any quantity of interest ⁇ with respect to the volumetric measure of the hole, i.e., in 2D:
- T j ⁇ ( p ) 4 1 + ⁇ ⁇ ⁇ ⁇ : ⁇ ⁇ ⁇ - 1 - 3 ⁇ ⁇ 1 - ⁇ 2 ⁇ tr ⁇ ( ⁇ ) ⁇ tr ⁇ ( ⁇ ) .
- topological sensitivity can be computed as follows: (1) finite element analysis (FEA) is carried over the domain, (2) stresses and strains are computed, and (3) then the topological sensitivity field is computed.
- FIG. 4C illustrates the resulting field 420 of topological sensitivity. The interpretation is that regions of low sensitivity correspond to regions with relatively lower impact on performance (and can be removed). Similar topological sensitivity fields can be computed for various performance metrics, both in 2D and 3D.
- FEA finite element analysis
- the PareTO method uses the topological sensitivity as a level-set to trace the Pareto curve for a decreasing-volume fraction. As the topology evolves, the topological sensitivity is recomputed at each iteration. Referring to FIG. 5A , for an intermediate topology 500 (of the first design 400 of FIG. 4A ), (1) FEA is carried over the topology 500 , (2) the stresses and strains are computed, and (3) the topological sensitivity field is computed through the above equation; the resulting topological sensitivity field 502 is illustrated in FIGS. 5B and 5C .
- the present methods modify the known PareTO methods to further have sensitivity for support volumes.
- An effective sensitivity field for support structure may take into consideration two metrics: (1) surface angle, and (2) overhang horizontal distance. Since these considerations may differ from one AM technology to the other, and comprehensive standards are yet to be devised, the present methods evaluate these criteria separately: first, a topological sensitivity is formulated based only on surface angle; then, this sensitivity formulation is modified to consider overhang horizontal distances.
- topological sensitivity for support structure volume may be evaluated as the rate of change in support structure volume with respect to volume metric measure of the hole.
- FIG. 6A illustrates a scenario where the exemplary design 300 of FIG. 3A is infinitesimally perturbed at a point P 1 in the interior of the topology. If a hole 602 of radius ⁇ is inserted in the interior of the domain ( ⁇ g ), the topological-shape sensitivity may be computed as follows. First, the topological derivative is computed via:
- T I S ⁇ ( p ⁇ ⁇ ) 3 ⁇ ( ⁇ - ⁇ ⁇ - sin ⁇ ( ⁇ ⁇ ) ⁇ cos ⁇ ( ⁇ ⁇ ) ) ⁇ ( sin ⁇ ( ⁇ ⁇ ) - sin 3 ⁇ ( ⁇ ⁇ ) 3 ) ⁇
- ⁇ /2 ⁇ circumflex over ( ) ⁇ is the threshold angle.
- FIG. 6B illustrates a scenario where the exemplary design 300 of FIG. 3A is infinitesimally perturbed at a point P 2 on the boundary of the topology.
- the support volume on the boundary depends both on the local neighborhood (curvature) and the length and direction of support.
- F S (x p ) at each boundary point we define a scalar function F S (x p ) at each boundary point as follows:
- the same sensitivity value is assigned to its corresponding opposite point.
- the above definitions give the support volume sensitivity at all points, illustrated by the sensitivity field 702 produced from the exemplary topology 500 .
- FDM fused deposition modeling
- the selection of r may be treated as a kernel smoothing operation that is build-direction dependent: at each overhang point, take the minimum subtended angle between all the neighboring boundary points that are closer than r and are underneath (or at least at the same layer) of the overhang point.
- the horizontal overhang distance is a type of kernel smoothing operation which is directional and assigns minimum value around a vicinity to its center.
- the original support-constrained TO problem above may be expressed in the standard form:
- ⁇ 0 ⁇ min ⁇ ( 1 , ( ⁇ - ⁇ ⁇ ⁇ g ) ) ⁇ - ⁇ ⁇ ⁇ g ⁇ 0 0 ⁇ - ⁇ ⁇ ⁇ g > 0 .
- the penalty parameter ⁇ is modified as follows:
- the resulting field 802 is a combination of the two fields 502 , 702 , and the relative weight is automatically determined from the Lagrangian formulation.
- the proposed method 900 for using the present TO framework to find an optimized isosurface, and therefore an optimized design, for a topology no of a given initial design proceeds as follows.
- a system executing the method 900 may solve the unconstrained optimization problem described above to obtain both the initial support volume S 0 of the initial object design, and the unconstrained support volume S unc. of the first unconstrained optimization of the initial design.
- it may be assumed that the unconstrained optimization problem has been solved, and the two parameters S 0 and S unc. have been computed and received by the system executing the method 900 .
- the system may compute the augmented (i.e., weighted) topological field T.
- the augmented topological field T includes: carrying out FEA on the topology ⁇ ; computing each of the normalized sensitivity fields T j , T s ; computing the weighted field T from T j and T s ; and smoothing the field T.
- FEA every time the topology ⁇ changes, FEA must be executed and the topological sensitivities recomputed.
- the system may extract a new topology ⁇ using fixed-point iteration.
- the system may compare the new topology to the previous topology to determine whether the topology has converged. If the topology has not converged, the system returns to step 904 to repeat the computations on the new topology. If the topology has converged, at step 910 the system may determine whether a desired volume for the isosurface has been reached. If so, at step 912 the system may output a final optimized isosurface corresponding to the last-extracted, converged topology. If the desired volume has not been reached, at step 914 the system may decrement the volume fraction and return to step 904 .
- FIG. 10A illustrates an exemplary 2D Messerschmidt-Bölkow-Blohm (MBB) beam 1000 , an object common to TO examples.
- the threshold angle is assumed to be 3 ⁇ /4.
- the initial design requires no support and the objective is to find the stiffest design at 0.65 volume fraction.
- FIG. 10B illustrates an unconstrained optimization 1010 performed without support volume constraints; the optimization 1010 presents a relative compliance of 1.29 (i.e., 129% of the initial MBB beam 1000 ), and its support volume is the baseline support volume against which the optimizations created by the present methods are compared.
- FIGS. 10C-E illustrate optimizations produced using the present support volume sensitive TO framework, with different support volume constraints applied. As expected, by reducing the desired support volume, the design is altered to reduce support volume accordingly. Unexpectedly, the alterations also result in increased compliance.
- the third constrained optimization 1040 requires 42% of the baseline support volume, and exhibits a relative compliance of 1.56.
- the triangular three-hole bracket 100 discussed above with respect to FIGS. 2 and 3B provides a suitable three-dimensional example.
- the threshold angle is assumed to be 3 ⁇ /4.
- the support volume is 0.79 cm 3 .
- the support volume is about 5 cm 3 and the relative compliance is 1.29.
- the objective is to find the stiffest design at 0.5 volume fraction.
- FIG. 11 shows an unconstrained curve 1102 and a constrained curve 1104 through 21 iterations of volume fraction reduction.
- the optimized isosurface 1130 exhibits a relative compliance of about 1.58.
- the optimized isosurface requires about 50% of the support volume required by the unconstrained solution. As theorized, removing more material can either increase or decrease the support volume due to its nonlinearity; nonetheless, imposing a stringent constraint on support structure consistently reduces the support volume.
- FIG. 13 in another example the present methods optimize a mount bracket 1300 .
- the threshold angle is again assumed to be 3 ⁇ /4.
- the build direction is selected to give the best surface quality on the larger cylindrical face 1302 .
- the support volume is 1.12 cm 3 .
- the objective is to find the stiffest design at 0.7 volume fraction.
- FIG. 14 illustrates the unconstrained optimized design 1400 .
- the final support structure volume for the unconstrained design 1400 is 9.24 cm 3 .
- the final support structure volume for the constrained design 1500 is 7.70 cm 3 , representing an approximately 17% reduction over the unconstrained design 1400 of FIG. 14 .
- FIG. 16 includes an unconstrained optimization curve 1602 and a constrained optimization curve 1604 illustrating the evolution of support volume throughout the optimization process. Up to 0.9 volume fraction the unconstrained and constrained results are very similar. However, for lower volume fractions the constrained support volume is consistently 20% smaller than that of the unconstrained design.
- FIG. 17 includes an unconstrained optimization curve 1702 and a constrained optimization curve 1704 illustrating the evolution of relative compliance values as more material is removed from the design.
- the final (C/C 0 ) is about 1.05, while by imposing the support constraint this value increases to about 2.52. This highlights the trade-off between support volume and compliance when the support constraint is imposed. It is essentially up to the designer to choose the intensity of the support constraint.
- each of these topologies was ‘printed’ on a XYZprinting Da Vinci 2.0 fused deposition printer.
- the support structures were not generated by the present methods, but introduced by the XYZprinting software, based on default settings.
- the three optimized designs have the same weight, as prescribed by the optimization, while the amount of support structures is substantially reduced in the constrained design. This example illustrates the effectiveness of the proposed methods in handling support constraints.
- the present TO framework and implementation methods are also robust with respect to different build directions of an object.
- FIGS. 18A-B the geometry of an automobile rocker arm 1800 is described via numerous curved surfaces and two cylindrical holes 1802 , 1804 in two different directions. This makes selecting the optimal build direction challenging. Further, to capture the complexity of the design, a hexahedral mesh with about 1.7 million degrees of freedom was used. A plausible choice for the build direction is one that provides for the larger cylinder 1806 to have better surface quality, or ⁇ Z in this case. Selecting the build direction substantially along the axis of the larger cylinder 1806 , as shown in FIG. 19 , has the further benefit for the unconstrained optimization 1902 that the initial support is minimal. Thus, the rocker arm design may be optimized for minimum compliance at 0.7 volume fraction without imposing any constraints on support structure.
- This disclosure provides a topology optimization framework that leads to designs with reduced support structures. Specifically, this disclosure introduces a novel topological sensitivity approach for constraining support structure volume during design optimization. The effectiveness of the proposed scheme was illustrated through several numerical examples, and demonstrated using FDM technology.
- Support structures were assumed to be vertical for simplicity, but the methodology can be extended to handle non-vertical support structures. Additionally, the weighting proposed herein is simple and easy to implement.
- the framework may include other AM-related constraints, such as surface roughness, volumetric error, inter-layer fusion, and so on.
- the proposed method may be coupled with methods for finding the optimum build direction to further reduce support volume.
- Table 1 shows that the improvements in object design for AM via the present support volume sensitive TO framework do not impose significant additional computational cost on the system (e.g., the computing device 100 of FIG. 1 ) generating the optimized design.
- the constrained problem requires more computational effort to compute support sensitivity field, yet for all of the presented experiments CPU time remains comparable.
- the information describes the exemplary optimizations presented herein, performed using a computing device with an 8-core Intel Core i7 CPU running at 3.00 GHz, 16 GB of memory, and the 64-bit version of the MICROSOFT WINDOWS 7 operating system.
Landscapes
- Engineering & Computer Science (AREA)
- Chemical & Material Sciences (AREA)
- Materials Engineering (AREA)
- Manufacturing & Machinery (AREA)
- Physics & Mathematics (AREA)
- Mechanical Engineering (AREA)
- Optics & Photonics (AREA)
- General Physics & Mathematics (AREA)
- Automation & Control Theory (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
- Moulds For Moulding Plastics Or The Like (AREA)
Abstract
Description
In the above equation, J=fTd is the compliance that must be minimized, |Ω0| is the initial design volume, Ω is the topology to be computed, and Vf is the desired volume fraction; K is the stiffness matrix, f is the external force vector, and d is the displacement vector.
If the performance metric is compliance, the field in 2-D is given by the closed-form expression:
In the above equation, S(Ωg) and V(Bg) are support volume and hole volume, for a hole of radius ε. Using the above definition, the support volume sensitivity is given by:
Where π/2≤α{circumflex over ( )}≤π is the threshold angle. For example, if the threshold angle α{circumflex over ( )}=π/2, then Ts(p)=1, i.e., the entire hole will need to be filled with support structures; a typical value is about 0.72, or 72% filled with support structures, when α{circumflex over ( )}=3π/4.
In the above equation, αp is the angle between surface normal and build direction at boundary point p. The sensitivity is computed for the worst-case scenario, where the boundary is perturbed along a support at each point s{circumflex over ( )}p. The sensitivity at the boundary is given by:
A popular method for solving such constrained optimization method is the augmented Lagrangian method, where the constraint and objective are combined to a single field, leading to an augmented topological field:
=(1−ωS)+ωS S
where the weight is defined via the augmented Lagrangian formulation:
μk÷1=max{μk −g,0}
The penalty parameter γ is modified as follows:
where typically ζ=0.25 and ζ=10.
TABLE 1 |
Computational Cost with and without Support Structure Constraint |
Finite element | CPU time | ||
degrees of | CPU time | Support | |
Example | freedom | Unconstrained | Constrained |
MBB | 27,400 | 5.25 sec. | 5.5 sec. |
Three-hole | 45,012 | 10 sec. | (η = 0.75) 11 sec. |
bracket | (η = 0.50) 13.7 sec. | ||
Mount bracket | 196,965 | 1 min 18 sec. | 1 min 29 sec. |
Rocker Arm (−Z) | ~1.7 million | 28 |
30 min 59 sec. |
Rocker Arm (+Y) | ~1.7 million | 28 |
32 |
Rocker Arm (+X) | ~1.7 million | 28 |
30 min 14 sec. |
Claims (20)
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
US15/269,264 US10613496B2 (en) | 2016-09-19 | 2016-09-19 | Support structure constrained topology optimization for additive manufacturing |
Applications Claiming Priority (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
US15/269,264 US10613496B2 (en) | 2016-09-19 | 2016-09-19 | Support structure constrained topology optimization for additive manufacturing |
Publications (2)
Publication Number | Publication Date |
---|---|
US20180079149A1 US20180079149A1 (en) | 2018-03-22 |
US10613496B2 true US10613496B2 (en) | 2020-04-07 |
Family
ID=61617873
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
US15/269,264 Active 2037-10-11 US10613496B2 (en) | 2016-09-19 | 2016-09-19 | Support structure constrained topology optimization for additive manufacturing |
Country Status (1)
Country | Link |
---|---|
US (1) | US10613496B2 (en) |
Cited By (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20200401102A1 (en) * | 2019-06-21 | 2020-12-24 | Qiang Cui | Frame Structure Optimization Method Based on 3D Printing |
US20220326683A1 (en) * | 2017-11-06 | 2022-10-13 | Abemis LLC | Method and system to generate three-dimensional meta-structure model of a workpiece |
Families Citing this family (22)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
CN107727189B (en) * | 2017-11-15 | 2020-01-14 | 珠海赛纳打印科技股份有限公司 | Method and device for acquiring structure volume, non-transitory computer readable storage medium and printer |
US10073440B1 (en) * | 2018-02-13 | 2018-09-11 | University Of Central Florida Research Foundation, Inc. | Method for the design and manufacture of composites having tunable physical properties |
US11914934B2 (en) | 2018-09-14 | 2024-02-27 | Siemens Industry Software Inc. | Active region adaptations for design domains in topology optimizations |
CN109741452B (en) * | 2019-01-10 | 2022-08-12 | 中南大学 | Automatic generation method of geological body 3D printing self-supporting structure |
WO2020204883A1 (en) * | 2019-03-29 | 2020-10-08 | Siemens Aktiengesellschaft | Method and system for optimizing process parameters in an additive manufacturing process |
US11014307B2 (en) * | 2019-05-17 | 2021-05-25 | Honeywell International Inc. | Method for generating and depicting additive manufacturing build supports |
US11308249B2 (en) | 2019-06-28 | 2022-04-19 | General Electric Company | Hybrid support structures for additively printed parts |
WO2021040705A1 (en) * | 2019-08-28 | 2021-03-04 | Siemens Aktiengesellschaft | Systems and method for processing topology optimized geometries |
US11221610B1 (en) * | 2019-09-30 | 2022-01-11 | General Electric Company | Optimized support design for sintering parts with complex features |
CN111444640B (en) * | 2019-11-15 | 2023-05-02 | 三峡大学 | Structural topology optimization method considering inclination constraint of additive manufacturing |
US11483942B2 (en) * | 2019-12-18 | 2022-10-25 | SpinLaunch Inc. | Ruggedized avionics for use on kinetically launched vehicles |
CN111079332B (en) * | 2019-12-18 | 2023-10-27 | 武汉联影智融医疗科技有限公司 | Design method of porous structure on surface of external fixing support |
CN111737839B (en) * | 2020-05-19 | 2023-03-31 | 广州大学 | BESO (beam-based event optimization) topology optimization method based on dynamic evolution rate and adaptive grid and application thereof |
US11675333B2 (en) * | 2020-06-26 | 2023-06-13 | Autodesk, Inc. | Generative design shape optimization with singularities and disconnection prevention for computer aided design and manufacturing |
CN112069715B (en) * | 2020-09-15 | 2022-09-20 | 吉林大学 | Topology optimization method based on multi-material structure |
CN112233242B (en) * | 2020-10-09 | 2022-08-05 | 西北工业大学 | Topological optimization design method of three-dimensional self-supporting structure |
CN112329163B (en) * | 2020-10-19 | 2022-09-06 | 南京理工大学 | Spacecraft support topological lattice bionic design method based on inherent characteristic constraint |
CN112395685B (en) * | 2020-11-06 | 2022-05-10 | 广州理工学院 | Topology optimization bicycle component design method suitable for additive manufacturing |
CN112685945B (en) * | 2021-01-11 | 2021-08-13 | 北京理工大学 | Magnetic-structure multi-physical-field topological optimization design method for additive manufacturing |
CN112765865B (en) * | 2021-02-04 | 2022-05-20 | 上海交通大学 | Support structure design method for controlling metal powder bed additive manufacturing thermal deformation |
CN113051796B (en) * | 2021-03-19 | 2022-10-21 | 湖南科技大学 | Structural topology optimization design method applied to additive manufacturing |
US11656602B2 (en) * | 2021-08-23 | 2023-05-23 | Palo Alto Research Center Incorporated | Physics-aware automatic spatial planning for subtractive and hybrid manufacturing |
Citations (7)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20060276925A1 (en) * | 2003-04-23 | 2006-12-07 | The Regents Of The University Of Michigan | Integrated global layout and local microstructure topology optimization approach for spinal cage design and fabrication |
US8355893B2 (en) * | 2008-12-12 | 2013-01-15 | Wisconsin Alumni Research Foundation | Method and system for analysis and shape optimization of physical structures using a computerized algebraic dual representation implicit dimensional reduction |
US20150190971A1 (en) * | 2014-01-09 | 2015-07-09 | Siemens Product Lifecycle Management Software Inc. | Method for structure preserving topology optimization of lattice structures for additive manufacturing |
US20150360288A1 (en) * | 2014-06-13 | 2015-12-17 | Zin Technologies, Inc. | Optimized additive manufacturing process |
US20170176975A1 (en) * | 2015-12-18 | 2017-06-22 | Dassault Systemes Simulia Corp. | Penalty function on design variables for designing variables for designing cost beneficially additive manufacturable structures |
US20170232515A1 (en) * | 2016-02-01 | 2017-08-17 | Seurat Technologies, Inc. | Additive Manufacturing Simulation System And Method |
US20170312986A1 (en) * | 2016-04-28 | 2017-11-02 | Wisconsin Alumni Research Foundation | Systems and methods for controlling support structures in manufacturing |
-
2016
- 2016-09-19 US US15/269,264 patent/US10613496B2/en active Active
Patent Citations (7)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20060276925A1 (en) * | 2003-04-23 | 2006-12-07 | The Regents Of The University Of Michigan | Integrated global layout and local microstructure topology optimization approach for spinal cage design and fabrication |
US8355893B2 (en) * | 2008-12-12 | 2013-01-15 | Wisconsin Alumni Research Foundation | Method and system for analysis and shape optimization of physical structures using a computerized algebraic dual representation implicit dimensional reduction |
US20150190971A1 (en) * | 2014-01-09 | 2015-07-09 | Siemens Product Lifecycle Management Software Inc. | Method for structure preserving topology optimization of lattice structures for additive manufacturing |
US20150360288A1 (en) * | 2014-06-13 | 2015-12-17 | Zin Technologies, Inc. | Optimized additive manufacturing process |
US20170176975A1 (en) * | 2015-12-18 | 2017-06-22 | Dassault Systemes Simulia Corp. | Penalty function on design variables for designing variables for designing cost beneficially additive manufacturable structures |
US20170232515A1 (en) * | 2016-02-01 | 2017-08-17 | Seurat Technologies, Inc. | Additive Manufacturing Simulation System And Method |
US20170312986A1 (en) * | 2016-04-28 | 2017-11-02 | Wisconsin Alumni Research Foundation | Systems and methods for controlling support structures in manufacturing |
Non-Patent Citations (62)
Title |
---|
A. A. Novotny et al., "Topological sensitivity analysis for three-dimensional linear elasticity problem," Computer Methods in Applied Mechanics and Engineering, Sep. 7, 2005, pp. 1-14. |
A. A. Novotny et al., "Topological-shape sensitivity method: Theory and Applications," Solid Mechanics and its Applications, 2006, 10 pages, vol. 137. |
A. Gebhardt, "Additive manufacturing design and strategies," Understanding Additive Manufacturing, Nov. 2011, pp. 103-128, Carl Hanser Verlag GmbH & Co. KG. |
A. Rietz, "Sufficiency of a finite exponent in SIMP (power law) methods," Structural and Multidisciplinary Optimization, Apr. 2001, pp. 159-163, vol. 21, Issue 2. |
A. S. Nezhad et al., "Pareto-based optimization of part orientation in stereolithography," Proceedings of the Institution of Mechanical Engineers, Part B, Journal of Engineering Manufacture, Oct. 2010, pp. 1591-1598, vol. 224, No. 10. |
A. T. Gaynor et al., "Topology optimization for additive manufacturing: Considering maximum overhang constraint," Presented at the 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Atlanta, GA, 2014, pp. 16-20. |
Carolyn Conner Seepersad et al. "A designers guide for dimensioning and tolerancing SLS parts," in Solid Freeform Fabrication Symposium, Austin, TX, 2012, pp. 921-931. |
Christopher B. Williams et al., "Design for additive manufacturing curriculum: A problem-and project-based approach," in International solid freeform fabrication symposium, Austin, TX, 2012, pp. 81-92. |
D. Brackett et al., "Topology optimization for additive manufacturing," in 22nd Annual International Solid Freeform Fabrication Symposium, Austin, TX, Aug. 17, 2011, pp. 348-362. |
Douglas S. Thomas et al., "Costs and Cost Effectiveness of Additive Manufacturing: A Literature Review and Discussion," NIST Special Publication 1176, Dec. 2014, 89 pages. URL: http://dx.doi.org/10.6028/NIST.SP.1176. |
E. Kesseler et al., "Multidisciplinary design analysis and multi-objective optimisation applied to aircraft wing," WSEAS Transactions on Systems and Control, ISSN 1991-8763, Dec. 2006, pp. 1-21, vol. 1, Issue 2, National Aerospace Laboratory NLR. |
Ercan M. Dede et al., "Topology optimization, additive layer manufacturing, and experimental testing of an air-cooled heat sink," Journal of Mechanical Design, DOI: 10.1115/1.4030989, Nov. 2015, pp. 111702-1-111702-9, vol. 137, American Society of Mechanical Engineers. |
Eric Barnett et al., "Weak Support Material Techniques for Alternative Additive Manufacturing Materials," Additive Manufacturing, Jun. 23, 2015, pp. 1-13. |
Feijóo, R. A. et al., "The Topological-Shape Sensitivity Method in two-dimensional linear elasticity topology design," Journal of Computational Methods in Sciences and Engineering, Applications of Computational Mechanics in Structures and Fluids, 2005, pp. 1-15, Cambridge International Science Publishing. |
G. K. Ananthasuresh et al., "A methodical approach to the design of compliant micromechanisms," Solid-State Sensor and Actuator Workshop, Jun. 13-16, 1994, pp. 189-192, Transducers Research Foundation. |
George I. N. Rozvany, "A critical review of established methods of structural topology optimization," Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-007-0217-0, Feb. 21, 2008, 21 pages, vol. 37, Issue 3, Springer-Verlag. |
Grégoire Allaire et al., "A level-set method for vibration and multiple loads structural optimization," Computer Methods in Applied Mechanics and Engineering,, Jun. 23, 2004, pp. 1-28. |
H. Lipson et al., "Fabricated: The New World of 3D Printing," 2013, John Wiley & Sons. |
Hans A. Eschenauer et al., "Topology optimization of continuum structures: A review*," Applied Mechanics Review, Jul. 2001, pp. 331-390, vol. 54, No. 4, American Society of Mechanical Engineers. |
I. Gibson et al., "Additive Manufacturing Technologies," 2010, Springer. |
I. Gibson et al., "Design Rules for Additive Manufacture," in International Solid Free Form Fabrication Symposium, Austin, TX, Aug. 9-11, 2010, pp. 705-716, University of Texas. |
Inna Turevsky et al., "Generalization of topological sensitivity and its application to defeaturing," ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, Sep. 4-7, 2007, pp. 335-344. |
Inna Turevsky et al., "Tracing the envelope of the objective-space in multi-objective topology optimization," ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington DC, USA, Aug. 28-31, 2011, pp. 805-813, American Society of Mechanical Engineers. |
J. J. Alonso et al., "Aircraft Design Optimization," Computational Science Research Center Computing Group, CSRCR2007-13, May 2007, 18 pages. |
J. Vanek et al., "Clever Support: Efficient Support Structure Generation for Digital Fabrication," Computer Graphics Forum, DOI: 10.1111/cgf.12437, 2014, pp. 117-125, vol. 33, No. 5, The Eurographics Association and John Wiley & Sons Ltd. |
Jan Sokolowski et al., "On Topological Derivative in Shape Optimization," Institut National de Recherche en Informatique et en Automatique, May 1997, pp. 1-31. |
Jérémie Dumas et al., "Bridging the Gap: Automated Steady Scaffoldings for 3D Printing," ACM Transactions on Graphics, Jul. 2014, 10 pages, vol. 33, No. 4, Article No. 98. |
Jianbin Du et al., "Topology optimization of continuum structures with respect to simple and multiple eigenfrequencies," Sixth World Congresses of Structural and Multidisciplinary Optimization, May 30-Jun. 3, 2005, pp. 1-9, Rio de Janeiro, Brazil. |
Jibin Zhao et al., "Determination of Optimal Build Orientation Based on Satisfactory Degree Theory for RPT," in Ninth International Conference on Computer Aided Design and Computer Graphics, Dec. 7-10, 2005, 16 pages. |
Jorge Nocedal et al., "Numerical optimization," Mathematics Subject Classification, Springer Science, 1999, 651 pages, Springer. |
Kailun Hu et al., "Support Slimming for Single Material Based Additive Manufacturing," Computer-Aided Design 65, Mar. 2, 2015, pp. 1-11. |
Krishnan Suresh et al., "Large-scale modal analysis on multi-core architectures," ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago IL, USA, Aug. 12-15, 2012, 7 pages, American Society of Mechanical Engineers. |
Krishnan Suresh et al., "Stress-constrained topology optimization: a topological level-set approach," Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-013-0899-4, Mar. 9, 2013, pp. 295-309, vol. 48, No. 2, Springer. |
Krishnan Suresh, "A 199-line Matlab code for Pareto-optimal tracing in topology optimization," Structural and Multidisciplinary Optimization, Nov. 2010, pp. 665-679, vol. 42, Issue 5. |
Krishnan Suresh, "Efficient Generation of Large-Scale Pareto-Optimal Topologies," Structural and Multidisciplinary Optimization, Jan. 2013, pp. 49-61, vol. 47, Issue 1. |
Krishnan Suresh, "Tracing Pareto-Optimal Frontiers in Topology Optimization," ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Montreal, Quebec, Canada, Aug. 15-18, 2010, 10 pages, American Society of Mechanical Engineers. |
Kurt Maute et al., "Level set topology optimization of printed active composites." Journal of Mechanical Design, DOI: 10.1115/1.4030994, Nov. 2015, pp. 111402-1-111402-13, vol. 137, American Society of Mechanical Engineers. |
L. Harzheim et al., "A review of optimization of cast parts using topology optimization II-Topology optimization with manufacturing constraints," Structural and Multidisciplinary Optimization, May 2006, pp. 388-399, vol. 31, No. 5. |
L. Wang et al., "Automobile body reinforcement by finite element optimization," Finite Elements in Analysis and Design, 2004, pp. 879-893, vol. 40, No. 8. |
Ligang Liu et al., "3D Printing Oriented Design: Geometry and Optimization" Siggraph Asia 2014 Course, SIGGRAPH Asia 2014 Shenzhen, Dec. 5, 2014, 49 pages. URL: http://staff.ustc.edu.cn/˜Igliu/Courses/SigAsia_2014_course_3Dprinting/index.html. |
M. Leary et al., "Optimal topology for additive manufacture: A method for enabling additive manufacture of support-free optimal structures," Materials & Design, Nov. 2014, pp. 678-690, vol. 63. |
M. Zhou et al., "The COC algorithm, Part II: topological, geometrical and generalized shape optimization," Computer Methods in Applied Mechanics and Engineering, Aug. 1991, pp. 309-336, vol. 89. |
Martin P. Bendsoe et al., "Topology Optimization-Theory, Methods, and Applications," Springer Science & Business Media, Oct. 7, 2003, pp. 1-6, Springer. |
Martin P. Bendsoe et al., "Topology Optimization—Theory, Methods, and Applications," Springer Science & Business Media, Oct. 7, 2003, pp. 1-6, Springer. |
Michael Yu Wang et al, "A level set method for structural topology optimization," Computer Methods in Applied Mechanics and Engineering, Jan. 2003, pp. 227-246, vol. 192. |
Ming Zhou et al., "Progress in topology optimization with manufacturing constraints," Ninth AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, GA, Sep. 4-6, 2002, 8 pages. |
Nobuyuki Umetani et al., "Cross-sectional Structural Analysis for 3D Printing Optimization," SIGGRAPH Asia, 2013, 4 pages, vol. 5. |
O. Sigmund, "A 99 line topology optimization code written in Matlab," Structural and Multidisciplinary Optimization, 2001, pp. 120-127, vol. 21, Springer-Verlag. |
P. M. Pandey et al., "Optimal part deposition orientation in FDM by using a multicriteria genetic algorithm," International Journal of Production Research, Oct. 2004, pp. 4069-4089, vol. 42, No. 19, Taylor and Francis Ltd. |
Paramita Das et al., "Optimum part build orientation in additive manufacturing for minimizing part errors and support structures," Procedia Manufacturing, doi: 10.1016/j.promfg.2015.09.041, 2015, pp. 343-354, vol. 1, Elsevier B.V. |
Qi Xia et al. "Simultaneous optimization of cast part and parting direction using level set method," Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-011-0690-3, Aug. 10, 2011, pp. 751-759, vol. 44, Springer-Verlag. |
Ratnadeep Paul et al., "Optimization of layered manufacturing process for reducing form errors with minimal support structures," Journal of Manufacturing Systems, Jul. 2015, pp. 231-243, vol. 36. |
Shinji Nishiwaki et al., "Topology optimization of compliant mechanisms using the homogenization method," International Journal for Numerical Methods in Engineering, Dec. 4, 1998, pp. 535-559, vol. 42, John Wiley & Sons, Ltd. |
T. E. Bruns et al., "Topology optimization of non-linear elastic structures and compliant mechanisms," Computer methods in applied mechanics and engineering, Mar. 16, 2001, pp. 3443-3459, vol. 190, No. 26-27. |
V. H. Coverstone-Carrol et al., "Optimal multi-objective low-thrust spacecraft trajectories," Computer Methods in Applied Mechanics and Engineering, 2000, pp. 387-402, vol. 186. |
W. E. Lorensen et al., "Marching cubes: A High Resolution 3D Surface Reconstruction Algorithm," ACM Computer Graphics, Jul. 1987, pp. 163-169, vol. 21, No. 4. |
Weiming Wang et al., "Cost-effective Printing of 3D Objects with Skin-Frame Structures," ACM Transactions on Graphics, Nov. 2013, pp. 1-10, vol. 32 No. 6, Article 177. |
X. Huang et al., "A new look at ESO and BESO optimization methods, " Structural and Multidisciplinary Optimization, DOI: 10.1007/s00158-007-0140-4, May 17, 2007, pp. 89-92, vol. 35, Springer-Verlag. |
X. Wang et al., "Structural shape and topology optimization in a level-set-based framework of region representation," Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-003-0363-y, May 2004, pp. 1-19, vol. 27. |
Xiaotin Zhang et al. "Perceptual Models of Preference in 3D Printing Direction," ACM Transactions on Graphics, Nov. 2015, pp. 1-12, vol. 34, No. 6, Article 215. |
Yuejun Jiang et al., "Solving problems with hard and soft constraints using a stochastic algorithm for MAX-SAT," 1st International Joint Workshop on Artificial Intelligence and Operations Research, Timberline, Oregon, 1995, pp. 1-15. |
Z. Luo et al., "Compliant mechanism design using multi-objective topology optimization scheme of continuum structures," Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-004-0512-y, Mar. 18, 2005, pp. 142-154, vol. 30, Springer-Verlag. |
Cited By (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20220326683A1 (en) * | 2017-11-06 | 2022-10-13 | Abemis LLC | Method and system to generate three-dimensional meta-structure model of a workpiece |
US20200401102A1 (en) * | 2019-06-21 | 2020-12-24 | Qiang Cui | Frame Structure Optimization Method Based on 3D Printing |
Also Published As
Publication number | Publication date |
---|---|
US20180079149A1 (en) | 2018-03-22 |
Similar Documents
Publication | Publication Date | Title |
---|---|---|
US10613496B2 (en) | Support structure constrained topology optimization for additive manufacturing | |
Mirzendehdel et al. | Support structure constrained topology optimization for additive manufacturing | |
Mullen et al. | Spectral conformal parameterization | |
Wang | A Quadtree-based adaptive Cartesian/Quad grid flow solver for Navier-Stokes equations | |
Jarkowski et al. | Towards consistent hybrid overset mesh methods for rotorcraft CFD | |
Dunning et al. | Coupled aerostructural topology optimization using a level set method for 3D aircraft wings | |
Poole et al. | Control point-based aerodynamic shape optimization applied to AIAA ADODG test cases | |
Gil et al. | An enhanced immersed structural potential method for fluid–structure interaction | |
Allen et al. | CFD-based optimization of hovering rotors using radial basis functions for shape parameterization and mesh deformation | |
Fogg et al. | Automatic generation of multiblock decompositions of surfaces | |
Liu et al. | A level-set-based topology and shape optimization method for continuum structure under geometric constraints | |
Chin et al. | A scalable framework for large-scale 3D multimaterial topology optimization with octree-based mesh adaptation | |
Oktay et al. | Three-dimensional structural topology optimization of aerial vehicles under aerodynamic loads | |
James et al. | An isoparametric approach to level set topology optimization using a body-fitted finite-element mesh | |
CN112989503A (en) | Designing 3D modeled objects by directional optimization | |
Lin et al. | Vertex-ball spring smoothing: an efficient method for unstructured dynamic hybrid meshes | |
Chen et al. | Tetrahedral mesh improvement by shell transformation | |
Zheng et al. | An improved local remeshing algorithm for moving boundary problems | |
Work et al. | Strand-grid-solution procedures for sharp corners | |
Geiss et al. | Topology optimization of active structures using a higher-order level-set-XFEM-density approach | |
Alauzet et al. | On the use of space filling curves for parallel anisotropic mesh adaptation | |
Liu et al. | A new layout optimization method for stiffened panels based on ground stiffener structure (GSS) and thickness penalty | |
Zahle | Wind turbine aerodynamics using an incompressible overset grid method | |
Gisbert et al. | Efficient implementation of flux reconstruction schemes for the simulation of compressible viscous flows on graphics processing unigs | |
Liu et al. | Automatic sizing functions for unstructured mesh generation revisited |
Legal Events
Date | Code | Title | Description |
---|---|---|---|
AS | Assignment |
Owner name: NATIONAL SCIENCE FOUNDATION, VIRGINIA Free format text: CONFIRMATORY LICENSE;ASSIGNOR:UNIVERSITY OF WISCONSIN, MADISON;REEL/FRAME:040261/0078 Effective date: 20161024 |
|
AS | Assignment |
Owner name: WISCONSIN ALUMNI RESEARCH FOUNDATION, WISCONSIN Free format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNORS:SURESH, KRISHNAN;MIRZENDEHDEL, AMIRMASSOUD;REEL/FRAME:041996/0040 Effective date: 20170210 |
|
STPP | Information on status: patent application and granting procedure in general |
Free format text: FINAL REJECTION MAILED |
|
STPP | Information on status: patent application and granting procedure in general |
Free format text: RESPONSE AFTER FINAL ACTION FORWARDED TO EXAMINER |
|
STPP | Information on status: patent application and granting procedure in general |
Free format text: ADVISORY ACTION MAILED |
|
STCV | Information on status: appeal procedure |
Free format text: NOTICE OF APPEAL FILED |
|
STPP | Information on status: patent application and granting procedure in general |
Free format text: DOCKETED NEW CASE - READY FOR EXAMINATION |
|
STPP | Information on status: patent application and granting procedure in general |
Free format text: NOTICE OF ALLOWANCE MAILED -- APPLICATION RECEIVED IN OFFICE OF PUBLICATIONS |
|
STPP | Information on status: patent application and granting procedure in general |
Free format text: PUBLICATIONS -- ISSUE FEE PAYMENT RECEIVED |
|
STCF | Information on status: patent grant |
Free format text: PATENTED CASE |
|
MAFP | Maintenance fee payment |
Free format text: PAYMENT OF MAINTENANCE FEE, 4TH YR, SMALL ENTITY (ORIGINAL EVENT CODE: M2551); ENTITY STATUS OF PATENT OWNER: SMALL ENTITY Year of fee payment: 4 |